COMPLEX MODULUS FROM INDIRECT TENSION TESTING

Transcription

1 COMPLEX MODULUS FROM INDIRECT TENSION TESTING By JAE SEUNG KIM A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ENGINEERING UNIVERSITY OF FLORIDA 22

2 ACKNOWLEDGMENTS I would like to thank my adviser, Dr. Reynaldo Roque. He always listened and respected my opinion. All tasks were accomplished under his support and guidance. I would like to offer heartfelt gratefulness and respect to him. I will never forget his help. I also would like to thank the other members of my committee, Dr. Bjorn Birgisson and Dr. Mang Tia. I appreciate their advice. I would like to thank Booil Kim, whose advice and knowledge helped me to complete my task. I would like to thank D. J. Swan for generous help. I also would like to thank Sungho Kim and Hong J. Kim for their friendship and encouragement. I wish to thank everyone in the Materials Group at University of Florida. Finally, I would like to thank my parents. They always gave me endless trust, encouragement, and support. ii

3 TABLE OF CONTENTS page ACKNOWLEDGMENTS...ii LIST OF TABLES... vi LIST OF FIGURES...vii ABSTRACT... ix CHAPTERS 1 INTRODUCTION Background Objectives Scope Approach LITERATURE REVIEW Indirect Tensile Test Superpave IDT Complex Modulus using Superpave IDT Review for Complex Modulus from Past Study Concept of Dynamic Modulus RILEM Report for Bituminous Binders and Mixes Witczak s Predictive Equation of Dynamic Modulus Hollow Cylinder Tensile Tester MATERIAL AND METHODS Specimen Preparation Equipment Test Procedure iii

4 4 DEVELOPMENT AND EVALUATION OF ANALYTICAL METHOD Introduction Conventional Analysis Method Regression Method Assessment of Regression Method Examination of Alternatives Development of Functions to Determine Properties from Complex Modulus Test Using Gaussian Kernel Window Smoothing Basic Concept of Gaussian Kernel Window Smoothing Development of a Function to Find Coefficient a Development of a Function to Find Maximum or Minimum Points Development of a Function to Determine Phase Angle and Magnitude between Rising Deformation Curve and Level Loading Curve Flow Diagram of Developed Functions Using Gaussian Kernel Window Smoothing Comparison between Conventional Regression and Peak Smoothing Comparison of Results from Computer-Simulated Data Preparation of Methodology for Comparing Analysis Results by Both Methods from Real Data Comparison of Results from Real Data Summary of Results SOFTWARE DEVELOPMENT Data Initialization Intermediate Calculation Final Calculation of Dynamic Properties Normalization Factors Trimmed Mean Deformation and Trimmed Mean Phase Angle Poisson s Ratio Calculated for Data Set Average Poisson s Ratio Correction Factors Horizontal and Vertical Moduli of E*, E and E Output of Program EVALUATION OF COMPLEX MODULUS DATA ANALYSIS PROGRAM Evaluation of Test Results Evaluation of Data Reduction Comparison between Normalized Dynamic Moduli CLOSURE Summary of Findings Conclusions Recommendations iv

5 APPENDIX A COMPARISION OF DATA ANALYSIS RESULTS BETWEEN TWO METHODS B OUTPUT OF.333 Hz C OUTPUT OF 1 Hz D OUTPUT OF 4 Hz E OUTPUT OF 8 Hz F DATA ANALYSIS RESULTS OF HORIZONTAL DYNAMIC PROPERTIES LIST OF REFERENCES BIOGRAPHICAL SKETCH v

6 LIST OF TABLES Table page 3-1. Mix design after wagoner (21) Test parameters of Superpave IDT Analysis results of both methods for computer-simulated data... 3 vi

7 LIST OF FIGURES Figure page 2-1. Dynamic modulus versus frequency after RILEM Report (1998) Dynamic modulus versus temperature after RILEM Report (1998) Dynamic modulus versus phase angle after RILEM Report (1998) Deformation curve from actual data Process of conventional regression method Typical loading signal at high frequency Non-symmetric sinusoidal curve simulation Basic concept of gaussian kernel window smoothing Concept of number of band widths Seven types of irregular data Determination of maximum or minimum Point Function to find maximum or minimum peak point Non-linear deformation problem Flow diagram of new developed method Generating computer-simulated data Problems associated with non-uniform loading or non-homogeneous material properties Comparison of E* from both methods at.333 Hz Comparison of E* from both methods at.5 Hz Comparison of E* from both methods at 1 Hz vii

8 4-17. Comparison of E* from both methods at 4 Hz Comparison of E* from both methods at 8 Hz Comparison of phase angle from both methods at.333 Hz Comparison of phase angle from both methods at.5 Hz Comparison of phase angle from both methods at 1 Hz Comparison of phase angle from both methods at 4 Hz Comparison of phase angle from both methods at 8 Hz Comparison of phase angle from both methods at all frequencies Comparison of dynamic modulus from both methods at all frequencies Horizontal phase angle Horizontal E* Horizontal E Horizontal E Normalized dynamic modulus viii

9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Engineering COMPLEX MODULUS FROM INDRECT TENSION TESTING Chair: Dr. Reynaldo Roque Cochair: Dr. Bjorn Birgisson Department: Civil and Costal Engineering By Jae S. Kim May 22 Due to its mechanistic characteristics, the complex modulus is replacing the current resilient modulus as new modulus to characterize asphalt mixture for design of interstate highway and most other high volume facilities. Although it would be of great practical value to obtain the complex modulus from indirect tension testing, the complex modulus has generally been obtained from uniaxial compression or tension tests. There is a clear need to develop and evaluate testing and analysis procedures to determine the complex modulus from indirect tension tests. The Superpave indirect tension test (IDT) was selected for this purpose. This test has been used successfully to measure creep compliance, tensile strength, and resilient modulus. The test and analysis procedures developed for the Superpave IDT were designed to overcome problems typically associated with conventional indirect tension testing systems. ix

10 Data reduction and interpretation procedures were developed and evaluated in this study to obtain complex modulus parameters (i.e., dynamic modulus and phase angle) from the Superpave IDT. It was determined that conventional regression methods can overestimate the true phase angle where asymmetric dynamic loads are applied. Asymmetric loading may sometimes be inevitable at higher loading frequencies, particularly when testing in indirect tension. A procedure that incorporates a gaussian kernel smoothing method was developed to overcome this potential problem. It was shown that complex modulus parameters determined with this method appeared to be accurate and exhibited appropriate trends with respect to loading frequency. In addition, values obtained from limited testing with the Superpave IDT appeared to be reasonable. Automated software was developed from analysis of complex modulus data from Superpave IDT tests. x

11 CHAPTER 1 INTRODUCTION 1.1 Background Since the 196s, many researchers have studied the complex modulus (dynamic modulus and phase angle) of asphalt mixtures. That is because the complex modulus has the potential of providing information and properties that are more suitable for mechanistic analysis and design of pavements. Many states are inclined to implement dynamic modulus for design and specification, but the test needs to be further refined. In NCHRP s (National Cooperative Highway Research Program) 22 Guide, the dynamic modulus was adopted as input level 1, which represents a design approach philosophy of the highest practically achievable reliability for asphalt concrete. The Asphalt Institute has adopted the dynamic modulus as the choice test to measure modulus. So, the dynamic modulus is being widely cited for characterizing asphalt concrete mixture in airfield and highway design procedures. Furthermore, AASHTO s (American Association of State Highway and Transportation Office) upcoming 22 Guide for design of pavement structures is planning to use the dynamic modulus to characterize mixture for interstate highway and most other high volume facilities. Therefore, it is important to understand all aspects of the complex modulus test, and at the same time, set up the proper testing and analysis methods. The indirect tensile test has been used successfully to measure the resilient modulus of asphalt concrete mixture. The critical stress location by load is generally considered to be at the bottom of the asphalt concrete layer and immediately underneath 1

12 2 the load, where the stress state is longitudinal and transverse tension combined with vertical compression. The stress state in the vicinity of the center of the face of an indirect tension specimen is very similar to this stress state. Consequently, the complex modulus obtained by indirect tensile test can be expected to provide reasonable complex modulus as well. 1.2 Objectives The objective of the work was to develop and evaluate analysis and data reduction procedures to obtain complex modulus (dynamic modulus and phase angle) using Superpave IDT. The specific objectives were To establish suitable data acquisition procedures to determine dynamic modulus, phase angle, E and E using the indirect tensile test. To identify appropriate data reduction procedures to determine dynamic modulus, phase angle, E and E using the indirect tensile test. To develop a computer program to determine the dynamic modulus, phase angle, E, E and Poisson s ratio automatically using Superpave IDT data. To conduct an evaluation of the data reduction software using actual test data. 1.3 Scope The complex modulus (dynamic modulus and phase angle) test using the Superpave system was first attempted at University of Florida, so many expected and unexpected problems had to be overcome. Therefore, only a limited amount of reliable laboratory test data was obtained for evaluation. However, this thesis focused on identifying suitable interpretation techniques that could address the numerous problems or sources of error that can occur when obtaining data from this test. Many, if met all of these problems can be simulated with actual data. So, the accuracy of the analysis method was verified with virtual data created to emulate observed and expected problems. In

13 3 addition, data from Superpave IDT tests performed at multiple frequencies were analyzed to evaluate the method and software developed. 1.4 Approach The steps involved in the research approach may be summarized as follows. Identify problems associated with the determination of complex modulus using Superpave IDT. Identify and evaluate appropriate analysis methods to determine complex modulus. Choose the best analysis method suitable for current Superpave IDT. Develop suitable data reduction and analysis procedures to determine complex modulus form IDT data.

14 CHAPTER 2 LITERATURE REVIEW 2.1 Indirect Tensile Test The indirect tensile test or diametral compression test is conducted by applying compressive loads with a haversine or other suitable waveform. The load is applied vertically along the vertical diametral plane of a cylindrical specimen of asphalt concrete. Roque and Ruth (1987) showed that modulus values determined by indirect tensile test resulted in excellent prediction of strain and deflection measured on full-scale pavements at low in-service temperatures (<3 C (86 F)) when the modulus was used in elastic layer analysis. At present, the Superpave IDT is used for determining the resilient modulus, tensile strength and creep compliance at lower temperature. 2.2 Superpave IDT The Superpave IDT was developed by Roque and Buttlar (1992), and Buttlar and Roque (1994) under the Strategic Highway Research Program. In their paper, they pointed out that accurate properties could not be obtained from the indirect tensile test without measuring accurate Poisson s ratio. In order to measure accurate Poisson s ration, Roque and Buttlar suggested two main recommendations. One was to revise the LVDT mounting system and the other was to consider the effect of specimen bulging effect through three - dimensional finite element analysis. The testing procedure and analysis equations resulting from their work are being used for the Superpave IDT, which is required for higher level mixture designs in Superpave. Their procedure is listed in AASHTO TP-9, Determining the creep compliance and strength of hot mix asphalt 4

15 5 (HMA) using the indirect tensile test device. Besides the creep compliance and strength, the Superpave IDT can be used to determine the resilient modulus of asphalt concrete. The resilient modulus from Superpave IDT results in more accurate and reasonable modulus values for asphalt concrete. Roque et al. (1997) completed the Superpave IDT software based on Roque and Buttlar (1992), and Buttlar and Roque (1994). They developed a trimmed mean approach to determine deformations of both faces, which enhanced accuracy for calculating resilient modulus using Superpave IDT. 2.3 Complex Modulus using Superpave IDT Since the 196 s, the complex modulus has been measured as a fundamental property by asphalt technologists (Papazian, 1962). As we know, bituminous materials display wide variations in properties depending on temperature, loading magnitude, loading time and loading frequency. As mentioned in Chapter 1, because of the relative complexity of the test, the dynamic modulus is generally considered to be more reasonable for evaluation and design of highway or high volume facilities. However, there are no acceptable testing or data interpretation methods to determine the complex modulus using the Superpave IDT. 2.4 Review for Complex Modulus from Past Study Concept of Dynamic Modulus In order to prevent any confusion, the complex modulus and the dynamic modulus have to be clearly discriminated. After performing the complex or dynamic modulus test, two factors are generally obtained. One is dynamic modulus, which is the ratio between stress and stain amplitudes, and the other is phase angle. The load typically applied in the complex modulus test is used as continuous sinusoidal. The sinusoidal loading function is expressed as σ = σ e iωt, in which σ is stress amplitude and ω is

16 6 frequency, which is the same as 2πƒ (where ƒ is cycles per second). At the same time, the reaction to the stress was expressed as strain: ε = ε e i(ωt-φ) in which ε is strain amplitude and φ is phase angle associated with damping of viscoelastic material. Therefore, the basic modulus equations are as follows: σ iφ E* = = E * e = E * (cosφ + i sinφ) iφ ε e E* = E' + i E" ( E' = E * cosφ, E" = E * sinφ) where : E*:Complex Modulus φ : Phase Angle E * : Dynamic Modulus E':Real Parts of Complex Modulus or Storage Modulus E": Imaginary Parts of Complex Modulus or Loss Modulus RILEM Report for Bituminous Binders and Mixes RILEM is one of the most active international research related organizations. They have done a lot of tests on Bituminous Binder and Mixture since Francken (1998) shows the approximate trend of complex modulus test parameters. Figure 2-1 to Figure 2-3 show typical relationships between dynamic modulus, phase angle, frequency and temperature. The examples presented in these figures were used for a standard French semi-granular bituminous mixture (5.4 % binder) in the two-point bending test on trapezoidal specimens in the uniaxial tension-compression test. Their research on complex modulus may be summarized as follows. Dynamic modulus decreases as temperature increases. Dynamic modulus increases as frequency increases. Phase angle decreases as frequency increases (for temperatures below 4 C). Phase angle increases as temperature increases (for temperatures below 4 C).

17 7 Figure 2-1. Dynamic modulus versus frequency after RILEM Report (1998) Figure 2-2. Dynamic modulus versus temperature after RILEM Report (1998)

18 8 Figure 2-3. Dynamic modulus versus phase angle after RILEM Report (1998) Witczak s Predictive Equation of Dynamic Modulus Witczak has performed a lot of dynamic modulus tests since He developed several predictive equations based on dynamic modulus test results. An equation developed in 1989 is representative of his work. The dynamic modulus tests (Witczak et al., 1989) were conducted on 149 separate asphalt mixes. The test method was uniaxial test such as triaxial compression without confining stress test as specified by ASTM D3497. Based on the equation (Witczak et al, 1989), the following conclusions can be obtained. Dynamic modulus decreases as temperature increases. Dynamic modulus increases as frequency increases Hollow Cylinder Tensile Tester Buttlar et al. (1999) developed a hollow cylinder tensile test for dynamic modulus of asphalt concrete. The basic principle of the test is to apply internal pressure to the inner cavity of a hollow cylinder specimen, which results in circumferential tension.

19 9 Applied stress is linearly related to applied pressure, and the resulting strain is linearly related to cavity volume change, which can be directly measured using strain gages. The reason for considering their test was to have an alternative to the IDT or direct tension test. Test results were well correlated with Witczak s predictive equation.

20 CHAPTER 3 MATERIAL AND METHODS The purpose of this work was to identify and overcome problems associated with the performance and interpretation of complex modulus tests performed with the Superpave IDT. This included the identification of a suitable data reduction and analysis method. The work also included development of computer software for automated data reduction and analysis. The system was used to evaluate the results of laboratory tests conducted by Wagoner (21). The sample and test methods used by Wagoner are summarized below. 3.1 Specimen Preparation The mixture preparation procedure is outlined in AASHTO T-283. First, the aggregates and asphalt binder were heated to 15 C (3 F) for three hours prior to mixing. Once the mixing was completed, the mixture was allowed to cool to room temperature for two hours. After the cooling period, the loose mixture was long term aged for 16 hours at 6 C (14 F). After the mixture was aged for 16 hours, it was reheated to 135 C (275 F) for two hours before compaction. The specimens were then compacted on the Superpave gyratory compactor. The compacted specimen size was 1 mm diameter by 1 mm high. The specimens were then cut by a wet saw for IDT testing. The mixture used in the complex modulus test from Superpave IDT is listed in Table 3-1 1

21 11 Table 3-1. Mix design after wagoner (21) Material type FDOT code Producer Pit No. Milled Material Top 5.45" N.B. & S.B. Roadway #7 Stone 52 Martine Maretta Aggregates GA-185 #89 Stone 51 Martine Maretta Aggregates GA-185 Anderson screens 2 Anderson Mining Corp Material Job mix formula Blend 15% 2% 4% 25% Sieve size 1" 1% 1% 1% 1% 1% 3/4" 1% 1% 1% 1% 1% 1/2" 1% 9% 1% 1% 98% 3/8" 1% 48% 99% 1% 89% 4" 84% 3% 29% 87% 47% 1 62% 2% 3% 65% 27% 4 39% 2% 2% 36% 16% 8 25% 1% 2% 19% 1% 2 9.6% 1% 1% 6% 4% SP.GR Optimum AC= 5.7% AC from milled material=.9% Recycling AC= 4.8% Gmm= 2.47 Note: Asphalt cement was AC Equipment The complex modulus test was conducted using Superpave IDT. The overall testing system includes MTS loading system, measurement, and data acquisition system. A brief description of each component follows: Loads were controlled using a MTS 81 Material Testing System. The data acquisition system used was Labtech Notebook Pro software. The measurements were obtained using extensometers designed by MTS specifically for use with the Superpave IDT. The method used to attach the extensometers to specimen was the same as that specified with Superpave IDT (Roque et al., 1997) a gage length of 1in was used for all specimens. All tests were performed at room temperature, so we did not use the temperature control system specified in Superpave IDT or AASHTO TP-9.

22 Test Procedure The complex modulus test was performed in load control mode by applying repeated and continuous sinusoidal load at a specified frequency. The load was selected to keep the horizontal strain in the linear viscoelastic range (typically below a horizontal strain of 35 µε) during test (Buttlar and Roque, 1994). The tested parameters are listed in Table 3-2. The procedures for indirect tensile complex modulus are summarized in the following steps: 1. Four brass gage points (5/16-inch diameter by 1/8-inch thick) were affixed with epoxy to each specimen face. 2. Extensometers were mounted on the specimen. Horizontal and vertical deformations were measured on each side of the specimen. 3. The test specimen was placed into the load frame. A seating load of 1 to 3 pounds was applied to the test specimen to ensure proper contact of the upper loading head. 4. The specimen was loaded by applying a repeated and continuous sinusoidal load to obtain horizontal strains below 35 µε. 5. When the applied load was determined, the computer software began recording test data. Table 3-2. Test parameters of Superpave IDT Frequency (Hz) Material SP-2 SP-2 SP-2 SP-2 SP-2 Temperature Room temp. Room temp. Room temp. Room temp. Room temp. Load (lbf) Thickness (in) Diameter (in)

23 CHAPTER 4 DEVELOPMENT AND EVALUATION OF ANALYTICAL METHOD 4.1 Introduction As indicated in Chapter 1, analysis of complex modulus data by hand has obvious limitations. Figure 4-1 shows that actual deformation measurements can have significant noise caused by electronic or mechanical vibration, the impact of loading force, or the reaction of the material. It is very hard to choose the maximum or minimum points of the deformation from these irregular curves. Therefore, the analysis is subjective and susceptible to user errors. To solve this problem, we absolutely need to set up a consistent data reduction and analysis procedure. Several methods were considered and evaluated. A summary of each method is presented in the following section. 8hz test (Example) Deformation(mils) Time(sec) Figure 4-1. Deformation curve from actual data 13

24 Conventional Analysis Method Regression Method One of the most popular methods used to interpret complex modulus data is the regression method. The regression method determines least-fit coefficients for a specified function by minimizing the least square error between the function and the measured data. Zhang et al. (1997) assumed the deformation data to follow a sine function superimposed on a linear function and the load data to follow a sine function. The deformation amplitude and phase angle were obtained as regression coefficients. It is conventional to find a phase angle and E* by the regression method. Figure 4-2 shows the process of finding the angle and amplitude. Figure 4-2. Process of conventional regression method

25 Assessment of Regression Method The merit of the regression method is that it is the most efficient method when an assumed equation and real data are well matched. However, errors increase as discrepancies between the assumed function and the real data increase. The regression method can be of particular concern in a sensitive test such as complex modulus. Figure 4-3 shows the measured loading shape when a testing frequency of 2 Hz was used in the Superpave IDT. In this case, the loading curve is not perfectly sinusoidal and since the deformation curve is determined by a loading curve, then it will also not be sinusoidal. It may be expected that tests performed on asphalt mixtures over a range of temperature, particularly with the Superpave IDT, may not result in perfect sinusoidal curves, especially at higher frequency. One reason is the inertia forces of the loading head increase as the loading frequency increases so, that the loading control system of testing machine may not be able to control the loading curve perfectly. 2Hz (random selection) 3 25 Load(lbf) Time Figure 4-3. Typical loading signal at high frequency

26 16 If the crooked sinusoidal curve were symmetric, there would be no problem in applying the regression method. However, as shown in Figure 4-3, the signal is not symmetrical. Therefore, the calculated value of phase angle is bigger than that of actual phase angle. This point can be easily made through computer simulation. In fact, a complete series of simulations were conducted to evaluate the effects of these types of errors on determination of complex modulus parameters using different methods. All computer simulations presented herein were performed by Mathcad. The signal in Figure 4-4 was generated by assuming the error in the sine wave was larger at the bottom than at the top of signal. Although it looks like a sine curve, it is not a sine curve. It was made by combining two quadratic functions. Both of the curves have the same function except for the phase angle, which was exactly 2, so the expected value of the phase angle was also 2. The table from Figure 4-4 shows the value of the phase angle found using the regression method was Therefore, it appears the regression method can result in significant errors for less than perfect data. Figure 4-4. Non-symmetric sinusoidal curve simulation

27 Examination of Alternatives To overcome this potential problem, two analysis methods were considered. One is the filtering method and the other is the smoothing method. In case of the filtering method, Fast Fourier Transform was chosen, and in case of the smoothing method, gaussian kernel smoothing method was chosen. The characteristics of both methods are described below. Fast fourier transform. FFT (Fast Fourier Transform) is based on DFT (Discrete Fourier Transform) algorithm. As reduced the number of computations of DFT, the DFT was improved as the FFT, which is much faster than the DFT. The FFT essentially decomposes or separates a waveform or function into sinusoids of different frequency. Consequently, extraneous frequencies of waveform or function can be eliminated using FFT. In spite of a lot of merits, FFT exposed many problems for analyzing complex modulus test data because FFT mainly used for signal analysis, so for complex modulus test, where it can be difficult to control even main the frequency, it resulted much greater problems than the regression method. Also, it is hard to discriminate real noise from the irregular deformation curve. In other words, some of noise may be true response, which should not be filtered. Gaussian kernel window smoothing. Gaussian kernel window smoothing uses local weighted averages of the vertical data to compute. This smoother function is most useful when data lies along a band of relatively constant width. Fortunately, the data from IDT have very constant width, and the result of application of this method showed very good quality. One of the major advantages of this method is that it can be applied to any curve including a curve resulting from an arbitrary function (i.e., having irregular data). However, this method also has problems. This method is not immediately suitable for

28 18 automated programming because a user has to determine the band width number for analysis. From evaluation of both methods, the gaussian kernel window smoothing method was determined to be a more suitable method to analyze the complex modulus test data. Although it has some problems, those problems can be overcome using programming skills. 4.4 Development of Functions to Determine Properties from Complex Modulus Test Using Gaussian Kernel Window Smoothing The following these steps were used to determine phase angle and dynamic modulus from complex modulus test data. 1. The gaussian kernel window smoothing method was applied to smooth irregular data or noise automatically and effectively from complex modulus test results. 2. Specific functions were developed to identify maximum or minimum data points from each cycle. 3. Specific functions were developed to accurately determine phase angle and amplitude from the maximum or minimum data points Basic Concept of Gaussian Kernel Window Smoothing As mentioned, the gaussian kernel window smoothing uses a local weighted average of input n-element vertical data to compute. A smooth function is only available when the horizontal data have a regular number band width. Once the number of band widths is determined, the gaussian kernel window smoothing function finds new vertical data, which depends on the number of band widths. Figure 4-5 shows the basic concept of gaussian kernel window smoothing method. The number of band width is represented by the symbol, b.

29 19 Figure 4-5. Basic concept of gaussian kernel window smoothing The key input required by the gaussian kernel window smoothing method is the number, a, which dictates how many ranges will be included when computing the gaussian kernel window smoothing function. According to the number, a, the result of gaussian kernel window smoothing will be changed. Figure 4-6 shows the result of using different input numbers. For each set of data, there is an optimal a that eliminates only noise and has minimal impact on the primary response. Lower or higher a values result in unsatisfactory smoothing curves.

30 2 Figure 4-6. Concept of number of band widths Development of a Function to Find Coefficient a The main problem associated with the implementation of the gaussina kernel window smoothing method is the proper selection of the coefficient, a. In order to define in order to define the optimal value of, a, the different types of irregular data observed during the complex modulus test must be analyzed. Figure 4-7 shows several types of irregular data observed during the complex modulus test. These seven types of irregular data were obtained from the results of hundreds of actual data sets.

31 21 Figure 4-7. Seven types of irregular data Special routines were developed to check for the presence of the seven types of irregular data. The idea is that additional smoothing is required as long as irregular data are found. So, the value of a is increased incrementally until the search routines find no irregular data in the signal. A computer routine was developed to accomplish this in a computationally efficient manner. Rather than checking the entire data set for irregular data at one time, overlapping groups of six data points are checked for irregular data, starting with the first six data points in the data set. Once the optimal value of a

32 22 determined for these first six points, the check is performed on the subsequent groups of six data points until no irregular data found in the entire data set. For example, data point numbers 1 through 6 are the first group checked, then data points 2 through 7 are the second group checked. This approach minimizes the number of checks that need to be made for irregular data. The steps involved in this automated gaussian kernel window may be summarized as follows: 1. Set a = Special routines check for presence of irregular data among the first of six data points. 3. If irregular data are found, then set a= a+1 and use gaussian kernel window function to smooth the data. 4. Repeat Steps 2 and 3 until no irregular data are found among the first group of six data points. 5. Repeat Steps 2 through 4 for subsequent groups of six data points until the end of the data set is reached. In the case of smoothing the loading curve, the data typically is either type 2 or type 3 because it is generally a smooth curve, and in the case of smoothing the deformation cure, all seven types can be encountered Development of a Function to Find Maximum or Minimum Points One of the merits of the gaussian kernel window smoothing is that can be applied to curves which have irregular data. However, function coefficients cannot be obtained directly from the gaussian kernel window smoothing function. It is simply a technique that arranges the irregular data smoothly. Another complexity is that the data acquisition system cannot obtain whole data points, in other words, when the data acquisition system typically record the test data at regular intervals. Consequently, there is generally some

33 23 difference between actual maximum or minimum data points and maximum or minimum data points obtained by simply smoothing the data. Figure 4-8 illustrates this point. Figure 4-8. Determination of maximum or minimum Point In order to find actual maximum or minimum data points, the regression method was used within a limited range. From the smoothed data, three maximum and minimum data points (see Figure 4-8) can be obtained from every cycle. The curve made up of the three data points is very small so that even if any functions were assumed, the difference in values obtained from any nonlinear functions is almost the same, as long as the assumed function passes through three maximum or minimum points. A quadratic function was assumed because the quadratic function can minimize time of computation. Figure 4-9 shows the procedure to find maximum or minimum point using limited

34 24 regression method assumed with a quadratic function. To find quadratic function coefficients, simple matrix and differential equation were used. Figure 4-9. Function to find maximum or minimum peak point

35 Development of a Function to Determine Phase Angle and Magnitude between Rising Deformation Curve and Level Loading Curve The phase angle between two level sine curves should be the same at top peak point as at bottom peak point. Also, the magnitude from level sine curve should be the same for the descending direction and rising direction. However, the phase angle between level sine curve and rising sine curve must be different at both top and bottom peak points, and also the magnitude of rising sine curve must be different between descending and rising directions. Figure 4-1 illustrates the problems. To find the correct phase angle between rising sine curve (deformation) and level sine curve (load) and the correct magnitude for the rising sine curve, at least two cycles are necessary. From the two cycles, one bottom phase angle and one top phase angle can be chosen. Also, one descending and rising magnitude can be chosen. Average angle of top and bottom phase angles and average magnitude of descending and rising magnitudes are exactly the same as the phase angle and magnitude from the conventional regression method as long as both curves are perfect sine curves. In summary: To determine phase angle and magnitude accurately, at least two cycles of loading curve and two cycles of deformation curve are necessary. The average of the top and bottom phase angles between loading curve and deformation curve is required to find the correct phase angle. The average magnitude of descending and rising deformation curves is required to find the correct magnitude. The complex modulus test data have many irregular data points or noise, so accurate values of phase angle and magnitude between loading curve and deformation curve cannot be expected with the functions above. The results of data analysis from the complex modulus test data indicated that the phase angle and magnitude between loading curve and deformation curve must be the average of several phase angles and magnitudes

36 26 from several cycles calculated using the functions above. The key in applying the functions below is the average value from too many cycles cannot be predicted on the change of phase angles and magnitudes and also, it is time consuming for calculation. Figure 4-1. Non-linear deformation problem Therefore, it is proper that five phase angles and magnitudes from 6 cycles calculated using the function below. A simple procedure to use the new function to determine phase angle and magnitude is as follows. To determine more accurate phase angle and magnitude, six cycles of loading curve and six cycles of deformation curve are necessary. 1. The average of five top phase angles and five bottom phase angles between loading curve and deformation curve from the six cycles is required to find accurate phase angle. 2. The average of five descending and rising magnitudes both loading curve and deformation curve from six cycles is required to find accurate magnitude.

37 27 3. Average of the averaged top and bottom phase angle is final phase angle. 4. Average of the averaged descending and rising magnitude is final magnitude Flow Diagram of Developed Functions Using Gaussian Kernel Window Smoothing Based on explanation above, the functions developed using gaussian kernel window smoothing method are described flow on the diagram presented in Figure Comparison between Conventional Regression and Peak Smoothing The newly developed method was named peak smoothing method. To confirm the accuracy of the peak smoothing method, two sets of data were used; computer simulated test data and real test data. An imaginary function composed of a sine function having a linear slope was used to generate the computer-simulated data. The two methods (conventional regression and peak smoothing) were applied to the same data range to evaluate accuracy and expected trends in both test data sets. In order to obtain the parameters, the following simple equations were used.

38 28 Figure Flow diagram of new developed method Comparison of Results from Computer-Simulated Data The deformation and load data from the complex modulus test were theoretically composed of sinusoidal curves. In case of the load data, a clean sine function was generated. In case of the deformation data, a sine function having a linear slope was generated. Additionally, the sine function having a linear slope was combined with another function to generate irregular data or noise. As illustrated in Figure 4-12, noise can be generated in either the horizontal or vertical direction. A function was used to generate random numbers. High random number means large noise. Figure 4-12 also shows the equations used to generate imaginary data. The following steps were used.

39 29 1. An imaginary load function composed of a sine function having constant height was generated. 2. An imaginary deformation function composed of a sine function having a linear slope, constant height, constant phase angle and randomly generated noise was generated. 3. An imaginary deformation curve was generated with a random number equal to zero (i.e., no randomly generated noise) 4. All data generated was analyzed using both methods: conventional regression and peak smoothing. 5. Compare and record the analysis results from both methods. 6. Repeat Steps 3 to 5 for increasing values of random numbers. The results of this effort are presented in Table 4-1. Figure Generating computer-simulated data

40 3 Table 4-1. Analysis results of both methods for computer-simulated data Phase Angle First try Second try Random Number Wishful Answer Smoothing Regression Smoothing Regression ~ ~ ~ : : -4~ : : -5~ : : -6~ : : -7~ : : -8~ : : -9~ : : Average Magnitude First try Second try Random Number Wishful Answer Smoothing Regression Smoothing Regression ~ ~ ~ : : -4~ : : -5~ : : -6~ : : -7~ : : -8~ : : -9~ : : Average Both methods showed good results so that the peak smoothing method appears to work well and to be as accurate as the conventional regression method Preparation of Methodology for Comparing Analysis Results by Both Methods from Real Data Problems associated with non-uniform loading or non-homogeneous material properties commonly occur in the Superpave IDT test specimens so that the deformation curves from each face are not complete sinusoidal curves having the same magnitudes. Consequently, the analysis results of conventional regression method and peak smoothing method are different because the analysis concepts of the two methods are different. Hence, the properties, phase angle, E*, E and E cannot be compared directly for each

41 31 face. However, based on computer simulation and literature review, the average of the properties such as phase angle, E*, E and E should be almost same as long as the load is perfectly sinusoidal. In other words, the average of deformations from both faces forms a complete sinusoidal curve when the loading curve was a sinusoidal curve. Figure 4-13 shows the results of computer simulation. Figure Problems associated with non-uniform loading or non-homogeneous material properties Comparison of Results from Real Data As stated earlier, the phase angles determined by the conventional regression method may not be accurate for testing Superpave IDT test results, because of the difficulties in achieving a symmetric loading signal. If the peak smoothing method could solve such a problem, then the Phase angles calculated by the two methods will be

42 32 different. Superpave IDT tests were performed at five loading frequencies:.33 Hz,.5 Hz, 1 Hz, 4 Hz, and 8 Hz. Phase angles and dynamic modulus were calculated using the average of the two horizontal deformation measurements. The same ranges were applied to both methods. Figure 4-14 to Figure 4-18 show the result of E* determined by each method and Figure 4-19 to Figure 4-23 show the result of phase angle analyzed by each methods. The results of each point were calculated from 1 cycles. As shown in Figure 4-14 and Figure 4-23, the dynamic moduli were almost the same for both methods at every frequency tested. However, the phase angles were not the same. As the frequency increased, the difference in phase angle between the two methods increased. It appears that the testing machine did not control loading frequency appropriately as the complex modulus test was performed at high frequencies. The testing machine generates an increasingly asymmetrical sinusoidal loading curve as the test frequency was increased..333hz 3 25 E*(psi) Time(sec) smoothing regression Figure Comparison of E* from both methods at.333 Hz

43 33.5hz Phase angle(degree) Time(sec) smoothing regression Figure Comparison of E* from both methods at.5 Hz 1hz E*(psi) Time(sec) smoothing regression Figure Comparison of E* from both methods at 1 Hz

44 34 4hz E*(psi) Time(sec) smoothing regression Figure Comparison of E* from both methods at 4 Hz 8hz E*(psi) Time(sec) smoothing regression Figure Comparison of E* from both methods at 8 Hz

45 35.333hz Phase angle(degree) Time(sec) smoothing regression Figure Comparison of phase angle from both methods at.333 Hz.5hz 35 3 Phase angle(degree) Time(sec) smoothing regression Figure Comparison of phase angle from both methods at.5 Hz

46 36 1hz Phase angle(degree) Time(sec) smoothing regression Figure 4-2. Comparison of phase angle from both methods at 1 Hz 4hz 3 Phase angle(degree) Time(sec) smoothing regression Figure Comparison of phase angle from both methods at 4 Hz

47 37 8hz Phase angle(degree) Time(sec) smoothng regression Figure Comparison of phase angle from both methods at 8 Hz 4.6 Summary of Results In the end, it appears that the phase angle cannot be accurately determined with the conventional regression method or any kind of regression method when the loading curve gets more crooked as the test frequencies increase. The peak smoothing method is relatively unaffected by mechanical or physical error of the testing machine and it uses almost the same procedure used in hand calculation. So it appears to be the best analysis method for the complex modulus test available at this time for the Superpave IDT. Figure 4-24 and 4-25 show the phase angle and dynamic modulus calculated by conventional regression method and peak smoothing method from horizontal measurements. The values at each frequency presented in Figure 4-24 and 4-25 were calculated by using a median function, which is a very useful function to eliminate error values as long as there is the sufficient number of data points. The median function finds the midpoint value for the data set. The individual values of phase Angle, E*, E and E is presented in Appendix A.

48 38 Smoothing V.S. Resgression Phas Angle (degree) Frequency (hz) Smoothing Regression Figure Comparison of phase angle from both methods at all frequencies Smoothing V.S. Regession Dynamic modulus (psi) Frequency (hz) Smoothing Regression Figure Comparison of dynamic modulus from both methods at all frequencies

49 CHAPTER 5 SOFTWARE DEVELOPMENT Based on Chapter 4, Roque and Buttlar (1992), and Roque et al (1997), the resilient modulus data analysis program was adopted for use with the complex modulus test. The procedure to find complex modulus was divided into three steps, because the complex modulus test deals with a large amount of data, and the three steps result in greater efficiency. The complex modulus tests from three replicate specimens were for 1 loading cycles on each specimen. Three data sets are required for analysis. The first step was to organize the raw data for analysis. The second step was to calculate phase lag (time difference between load and strain) and the magnitude of load and deformation. The last step was to determine final phase angle, E*, E and E based on results from the three specimens. 5.1 Data Initialization The complex (dynamic) modulus computer data analysis program (ITLT_dynamic) requires data in an Excel file, which means the original data obtained from data acquisition system must be placed in an Excel file. Also, 1 cycles loading data were used for each test. The purpose of this step was to change the raw data obtained from the data acquisition system to manageable data. The raw deformation data included initial vibration before loading and were not set to zero. Therefore, the data initialization step accomplishes three things: It eliminates the data obtained prior to loading. It calculates the absolute change in deformations relative to the start of loading. It stars their data in an Excel file. 39

50 4 5.2 Intermediate Calculation In this step, the peak smoothing method was used to find basic properties such as phase angle (time difference between peak load and peak strain), magnitude of load and magnitude of deformation. The complex modulus test results vary from cycle to cycle, so to measure the basic properties such as phase angles and the magnitudes, just one or two cycles are not enough. As specified in Chapter 4, six cycles were selected for analysis, from which five values of top phase angles, five values of bottom phase angles, and five values of descending and rising magnitudes are obtained. The complex (dynamic) modulus data analysis program regards the range, six cycles, as one set. Consequently, from the one set, the peak smoothing method is applied to each set. The complex (dynamic) modulus data analysis program requires 1 cycles for analysis. Among the 1 cycles of tested data, analyses were performed to obtain parameters every 1 cycles so that a total 1 analyses were performed at each frequency. The intermediate calculation consists of the following steps. 1. Consider six cycles as one data set. 2. Obtain 1 data sets from the 1 cycles of data obtained (i.e., one data set every 1 cycles). 3. Calculate 5 values of top phase angles, bottom phase angles, descending magnitudes and rising magnitudes of loading curve and deformation curve from each data set. 4. Calculate average values from the five values obtained. 5. Repeat Steps 4 and 5 for each data set. 6. The Intermediate calculation resulted in ten sets of values obtained from cycle number 1 to cycles Final Calculation of Dynamic Properties This step requires data results obtained from Step1 (data initialization) and Step 2 (intermediate calculation). In addition, three data sets obtained from three replicate

51 41 specimens are required for proper interpretation using the Superpave IDT. The horizontal deformation carries the symbol, H, while the vertical deformation carries the symbol, V. The horizontal phase angle carries the symbol, PA_H, while the vertical phase angle carries the symbol, PA_V Normalization Factors Since different specimens may have different thickness, diameter, or load, the deformations need to be normalized. Buttlar and Roque (1994) developed the following equations, Equation 5.1, to get normalization factors. 3 t i t = i = 1 AVG 3 3 D i D = i = 1 AVG 3 3 P i P = i = 1 AVG 3 t D P C = ( i ) ( i ) ( AVG ) NORM i t D P AVG AVG i ΔH = ΔH C NORM j j NORM i ΔV NORM = ΔV C j j NORM i